#355126

order	512
nilpotency class	3
isoclinism	isoclinic class of #17405
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1438 conjugacy classes of subgroups; only the 402 normal subgroups are shown.

normal subgroups only

This group has 1438 conjugacy classes of subgroups — too many to draw. Only its 402 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

(5,6)(7,10)(8,9)(11,12)(13,14,17,23,18,22,16,21)(15,19,24,27,25,20,26,28);(1,2,3,4)(6,9)(7,11)(13,15)(14,19)(16,26)(17,24)(18,25)(20,22)(21,28)(23,27);(5,7,8,11)(6,10,9,12)(13,15)(14,20)(16,24)(17,26)(18,25)(19,22)(21,28)(23,27);(6,9)(10,12)(13,16,18,17)(14,21,22,23)(15,24,25,26)(19,27,20,28)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=ce,d^b=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262322);